The Dynamics of Viruslike Capsid Assembly and Disassembly

Cowpea chlorotic mottle virus (CCMV) is a widely used model for virus replication studies. A major challenge lies in distinguishing between the roles of the interaction between coat proteins and that between the coat proteins and the viral RNA in assembly and disassembly processes. Here, we report on the spontaneous and reversible size conversion of the empty capsids of a CCMV capsid protein functionalized with a hydrophobic elastin-like polypeptide which occurs following a pH jump. We monitor the concentrations of T = 3 and T = 1 capsids as a function of time and show that the time evolution of the conversion from one T number to another is not symmetric: The conversion from T = 1 to T = 3 is a factor of 10 slower than that of T = 3 to T = 1. We explain our experimental findings using a simple model based on classical nucleation theory applied to virus capsids, in which we account for the change in the free protein concentration, as the different types of shells assemble and disassemble by shedding or absorbing single protein subunits. As far as we are aware, this is the first study confirming that both the assembly and disassembly of viruslike shells can be explained through classical nucleation theory, reproducing quantitatively results from time-resolved experiments

Optimization of the SEC protocol for studying the self-assembly dynamics during size reduction S10 3 Theoretical methods and discussions S12 3.1 Theory S12 3.2 Numerics S17 3. 3 The fraction of protein dimers in free solution and in capsids S18 3.4 Fitting of the simulation results with respect to different reference points S19 S 2 3.5 1 Experimental materials and methods

Materials
Ampicillin, chloramphenicol, yeast extract, and peptone were purchased from Sigma-Aldrich/ Merck. Isopropyl-β-D-thiogalactopyranoside (IPTG) was obtained from PanReac AppliChem VWR. Ni-NTA agarose beads were obtained from Qiagen.  All buffers were filtered over a 0.2 micron filter prior to use.

UV-vis absorbance measurements
In order to determine the protein concentrations during experiments, the absorbance at 280 nm was measured with a spectrophotometer ND-1000 and the concentrations were subsequently calculated using the theoretical extinction coefficients. 1

Mass spectrometry
Protein mass characterization was performed using a High Resolution LC

Size exclusion chromatography (SEC)
SEC analysis was performed on a Superose 6 increase 10/300 GL column (GE Healthcare Life Sciences). Analytical measurements were executed on an Agilent 1260 bio-inert HPLC. Samples with a protein concentration of 100 µM were separated on the column at 21 °C with a flow rate of 0.5 mL/min. Running buffer was "pH 7.5, 100 mM NaCl no EDTA buffer" for T = 3 to T = 1 shift and "pH 5.0 buffer" for T = 1 to T = 3 shift (see table S1 for exact compositions).

Transmission electron microscopy (TEM)
TEM grids (FCF-200-Cu, EMS) were glow-discharged using a Cressington 206 carbon coater and power unit. Protein samples (10 µM, 5 µL) were applied on the glow-discharged grids and incubated for 1 min. The samples were carefully removed using a filter paper. Then, the grid was negatively stained by applying 2% uranyl acetate in water (5 µL). The staining solution was removed after 15 seconds and the grid was allowed to dry for at least 15 minutes. The samples were studied on a FEI Tecnai 20 (type Sphera) (operated at 200 kV, equipped with a LaB6 filament and a FEI BM-Ceta CCD camera).

Dynamic light scattering (DLS) measurements
DLS measurements were performed on a Malvern Zetasizer Nano ZSP at 21°C, unless stated otherwise. Samples (100 µM, unless stated otherwise) were centrifuged twice prior to analysis.
Buffers were filtered prior to use. All measurements were done in triplo, and the average of the triplo measurements was plotted.

General protocol for measuring the self-assembly dynamics during size increase (conversion dynamics from T=1 to T=3 particles)
For a typical dynamics experiment a 100 µM VW1-VW8 ELP-CCMV coat protein solution (150 µL -1200 µL) in pH 7.5 buffer with 500 mM NaCl was prepared and dialyzed to pH 5.0 buffer at 4 °C (12-14 kDa MWCO). Dialysis buffer (150 mL -200 mL) was changed after 30 minutes and 60 minutes. At different time points during dialysis, 110 µL and/or 5 µL samples were retrieved from the dialysis membrane, spun down for 1 minute at 13400 rpm, and subjected to SEC analysis and/ or TEM analysis respectively.
Hereafter, the mixtures were dialyzed at 4 °C either to pH 5.0 buffer (T=3 capsids) or to pH 7.5 buffer, 500 mM NaCl (T=1 capsids) and incubated in the final buffer at 4 °C for up to one week.

S 7
At intermediate time points, samples were taken, heated to 21 °C with 1 °C/min and subjected to SEC analysis with pH 5.0 buffer or pH 7.5 buffer with500 mM NaCl as eluent. At the final time point, fractions were collected during a preparative SEC run and the combined capsid fractions were analyzed with SDS-PAGE and TEM. The amount of mEGFP incorporated into the capsids was determined by SDS-PAGE analysis. Gels that were visualized via Coomassie Brilliant Blue staining (Biorad) were analyzed with ImageJ gel analysis software to calculate the loading of capsids with mEGPF. Hereto, the following formula was used: where gel is the intensity of the protein band on the SDS-PAGE gel as determined by ImageJ analysis; mw is the molecular weight of the protein.

Optimization of conditions to allow for optimal dynamics
In order to study VW1-VW8 ELP-CCMV capsid size shifts, we first investigated the optimal conditions that would allow for the dynamic behavior of the capsids. In preliminary results we observed a size shift from T = 1 capsids to T = 3 capsids during overnight dialysis from pH 7.5 to pH 5.0 at 4 °C ( Figure S1). We, therefore, evaluated with SEC whether the reverse size shift would also take place. We observed that only a partial shift from T = 3 to T = 1 capsids took place during the overnight dialysis to pH 7.5 buffer with 100 mM NaCl ( Figure S1A,B first two chromatograms in each panel). We then proceeded by investigating whether a second overnight dialysis to pH 5.0 would induce a re-shift back to T = 3 capsids as well. Interestingly, this was the case only when the dialysis was performed at 4 °C ( Figure S1B), while at 21 °C no size shift was observed. This indicates that at 21 °C the capsids are much less dynamic than at 4 °C. Although this seems contra-intuitive at first, this observation can be explained by the interactions between the hydrophobic ELP-domains which are much stronger at 21 °C than at 4 °C. So it is highly likely that these interactions in the capsid interior hamper rearrangements of the CP domains in the capsid shell, which are necessary for a size shift.
Another factor that was thought to influence the capsid dynamics as a result of ELP interactions is the ionic strength of the buffers used. Previously, we used a pH 7.5 buffer with 100 mM NaCl to completely disassemble other ELP-CCMV variants. Therefore, we hypothesized that this ionic strength would also allow for dynamics within capsids of our more hydrophobic VW1-VW8 ELP-S 8 CCMV variant. As, ideally, we wanted to study the capsid size shifts while varying as few factors as possible, preferably only the pH, we attempted to store T = 3 capsids in pH 5.0 buffer with 100 mM NaCl instead of 500 mM NaCl. However, this, unfortunately, led to the aggregation of the protein already within 16 hours (data not shown). We, therefore, evaluated whether VW1-VW8 ELP-CCMV exhibited dynamic behavior when dialyzed from pH 5.0 buffer with 500 mM NaCl to pH 7.5 buffer with 500 mM NaCl, thus only changing the pH. Unfortunately, only a very small part of the capsids appeared to be shifted in size after 24 hours ( Figure S1C) as compared to dialysis to pH 7.5 buffer with 100 mM NaCl ( Figure S1D), which can again be explained by hydrophobic interactions between ELP domains hampering capsid dynamics at 500 mM NaCl. We, therefore, decided to use a shift from pH 5.0 with 500 mM NaCl to pH 7.5 with 100 mM NaCl and vice versa to study the dynamics during VW1-VW8 ELP-CCMV capsid size decrease and increase respectively as is discussed further in the main text.

Optimization of dialysis conditions
As described in the previous section, it is necessary to both change the pH and NaCl concentration in order to study size shifts of VW1-VW8 ELP-CCMV capsids. If only the pH would have to be adjusted, this could have been done by adding either HCl or NaOH to the buffer.
However, to also adjust the NaCl concentration a buffer exchange step is necessary. Although spin-filtration would be the quickest option and would allow for evaluation of the capsid size upon change of the conditions very quickly, it would also introduce changes in the protein concentration, which could affect the capsid assembly state. As this could complicate our evaluation of capsid size changes as a function of pH, we decided to perform dialysis in order to change the pH and NaCl concentration, despite being a slower process than spin-filtration.
During initial experiments, it was observed that there is a large dependency of capsid dynamics on dialysis time while incubation periods at 4 °C were kept constant ( Figure S2A). When a T = 3 capsid solution was dialyzed to pH 7.5 buffer with 100 mM NaCl, a large shift to T = 1 capsids was only observed during SEC analysis when the total dialysis time was more than 30 minutes.
This indicated that either the pH or the NaCl concentration changed slower than anticipated during 30 and 60 minutes and the dialysis buffer was either stirred at 150 rpm (blue circles) or not stirred (yellow squares). All measurements were performed in triplicate and data is presented as mean ± standard deviation.
S 10 dialysis. As the pH switch during dialysis takes place within minutes, it was suspected that the other variable during dialysis, the NaCl concentration, changed more slowly. The amount of NaCl that is dissolved in an aqueous solution affects the conductivity of that solution, thus conductivity measurements were performed to follow the change of the NaCl concentration during dialysis.
Hereto, a mock dialysis with the same ratio between the volume inside the dialysis bag and the solvent volume was performed and the conductivity was monitored over a time course of 4 hours ( Figure S2B). A dialysis time of around 2 hours was necessary to fully convert the NaCl concentration from 500 mM to 100 mM NaCl in the dialysis bag, which explains why dialysis time is such an important determinant of capsid dynamics.

Optimization of the SEC protocol for studying the self-assembly dynamics during size reduction
As during initial dynamics experiments large quantities of dimers were observed in the SEC chromatograms when pH 7.5 buffer with 100 mM NaCl was the eluent, while these were never observed for VW1-VW8 ELP-CCMV before, the origin of these dimers was evaluated. Hereto, a 100 µM VW1-VW8 ELP-CCMV coat protein solution in pH 5.0 buffer was dialyzed (MWCO 12-14 kDa) to pH 7.5 buffer with 100 mM NaCl at 4 °C overnight and subsequently spiked with known amounts of native ELP-CCMV dimers in the same buffer. DLS and native PAGE were employed to analyze the capsid-dimer mixtures. From the DLS results in Figure S3A and B, it becomes clear that DLS is not sensitive enough to detect dimers in capsid-dimer mixtures, which could be explained by the high scattering of the capsids overpowering any light scattering caused by the much smaller dimers. Therefore, although no dimers are detected with DLS of the dialyzed capsids, this does not confirm that indeed no dimers are present in this capsid solution. We therefore focused on native PAGE analysis. From the results in Figure S3C, it can be appreciated that capsids and dimers can be easily distinguished from each other. Furthermore, based on the band intensities on the gel it can be stated that in the VW1-VW8 ELP-CCMV capsid solution less than 5 % dimers are present. This indicates that the large fractions of dimers that are observed in the SEC chromatogram are most likely the result of some disassembly of VW1-VW8 ELP-CCMV capsids taking place during SEC analysis, which might be caused by the extreme dilution (240 times) during the chromatographic procedure.

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As for data analysis purposes, the SEC chromatograms should provide the best representation of the assembly state of VW1-VW8 ELP-CCMV during dynamics, a protocol was developed for inhibiting capsid dynamics prior to SEC analysis. Hereto, a 100 µM VW1-VW8 ELP-CCMV solution in pH 5.0 buffer was dialyzed (MWCO 12-14 kDa) to pH 7.5 buffer with 100 mM NaCl at 4 °C overnight and subsequently incubated with various amounts of NiCl2 for 50 minutes (the duration of one SEC run) at room temperature. 6 From the results in Figure S4A and B, it can be observed that the addition of at least 0.2 equivalents Ni 2+ (relative to the amount of VW1-VW8 ELP-CCMV coat protein concentration) successfully reduces the number of dimers that are observed in the SEC chromatograms, without affecting the T = 3 : T = 1 ratios. With increasing amounts of Ni 2+ also some higher-order structures became visible on the SEC chromatograms (around 7 mL), indicating that high amounts of Ni 2+ alter the protein fractions in the VW1-VW8 ELP-CCMV solution. Therefore, the addition of 0.2 equivalents of Ni 2+ was found to be most suitable to reduce the number of dimers introduced due to dilution on the SEC column while not affecting the protein fractions. To confirm that the addition of this amount of Ni 2+ effectively stops any dynamics, a sample that was dialyzed for 30 minutes was subjected to SEC analysis after 50 minutes or 7 hours of incubation with Ni 2+ (Figure S4C,D). The resulting SEC traces and protein fractions were very similar, indicating that the addition of 0.2 equivalents of Ni 2+ effectively stops the dynamics and stabilizes the samples for prolonged storage prior to SEC analysis.

Theory
Based on previous experimental, theoretical and computer simulation results, we put forward that nucleation is the underlying mechanism for capsid assembly and disassembly. [7][8][9] To this end, we combine equilibrium theory, borrowed from the physics of supramolecular polymers, and classical nucleation theory. 10 This allows us to calculate the time evolution of the assembly and disassembly of mixtures of capsids, the predictions of which we compare with our experimental results. Note that the assembly kinetics of free subunits into competing capsids with different numbers has been discussed before 11,12 -a kinetic theory of T number conversion has not yet been attempted.

S 13
To obtain the thermodynamic parameters required for our kinetic theory, we write the free energy of an aqueous solution in which only free ELP-CCMV subunits and the fully formed capsids are allowed to be present. In the equilibrium theory we ignore the intermediate states, as previous experimental and simulation work, as well as the findings presented in this paper clearly show that they are barely detectable (if at all) and represent short-lived states. 13 i.e., can never exceed * 1 = 1 or * 3 = 3 . Thus, for each type of capsid, there is a critical free protein concentration * below which the concentration of capsids is almost zero as the number of subunits in the capsids, , is large compared to unity. For CCMV, the basic protein subunits are dimers, so 1 = 30 for the = 1 and 3 = 90 for the = 3 capsid.

S 14
Using the equilibrium theory described above, we can now set up the kinetic theory of capsid assembly and disassembly within the framework of CNT. 8 The Gibbs free energy of the formation of an incomplete spherical capsid of the species containing = 1, . . . , molecules with a circular rim can be written as / is a dimensionless magnitude of the rim energy, with the radius of the shell and the free energy cost per unit length of the rim. 10 can be estimated as = − where ∈ [0,1] is a geometric factor indicating the average fraction of bonds that a subunit on the rim is missing, which depends on the local coordination number and roughness of the rim. is the effective diameter of a protein unit that is approximated as a disk. Assuming that the surface of a fully formed capsid is covered entirely by capsid proteins, the effective diameter can be written as = ( 2 / 2 ) = * = √ (1 + 2 ) 3/4 is the so-called Zeldovich factor that describes the sharpness of the free energy barrier and that may be interpreted as a measure of the lifetime of the critical nucleus of size * . 13 The attempt or attachment frequency * of the monomers attaching to the critical nucleus depends on the mode of attachment, and may be , e.g., a function of the diffusivity and concentration of the free monomers, the size of critical nucleus, and on some internal molecular time scale associated with the docking process that may depend on conformational switching. 20 For simplicity, we assume that it does not depend on the size of the clusters nor on the concentration.
To model the disassembly process, we presume that the initial state constitutes a fully formed where , * represents the free energy barrier for the disassembly of a shell to form monomers.
Notice that the dissociation rate depends on , the capsid concentrations of species = 1,3. We shall presume that the attachment frequency associated with the association process is the same as that of the dissociation process, as it describes the same process and we presume it to be independent of the size of the critical nucleus. 21 Because capsids with different numbers have different radius of curvature, we do not allow for a direct transition from one number to another one. In our allowed reaction path pathway, growth or disassembly can only proceed by the shedding or docking of individual protein subunits, which for CCMV constitute coat protein dimers. This is not a far-fetched reaction path, as our experiments show no indication of partially disassembled = 3 particles spontaneously morphing into = 1 particles, or vice versa, = 1 particles opening up to absorb subunits and growing into a = 3 particles. Hence, we presume that, first, one type of capsid disassembles into dimers, second, free dimers reassemble into different capsid sizes following their corresponding assembly nucleation rates.
Presuming that kinetic processes are sufficiently slow to allow us to use the expressions for steady-state nucleation rates for association and dissociation, i.e., presume a quasi steady state It is important to realize that the time that it takes to change the and the salt concentration of the buffer solution might not be exactly the same in each experiment. In addition, the lag time for assembly and disassembly of capsids with different sizes are arguably different. Therefore it is difficult to pinpoint the actual "time zero" for each individual experiment. In order to deal with this uncertainty, we start collecting data 30 minutes after the experiment commences. We also assume that the lag times are negligible on the time scale of the experiment, thus we ignore the S 17 first phase of assembly in CNT in which capsids have not started to assemble or disassemble. 10

Numerics
The kinetics equations predicted by CNT (Eqs. S7 and S8) are solved by using finite difference methods. Assembly and disassembly nucleation rates at the beginning of the simulation are determined by the initial conditions. The concentrations of capsids and free dimers are calculated at each time step, using the values and nucleation rates at the previous time step. Hence, our time-stepping equations read: Eq. (S12) where = | , − , | is the absolute assembly or disassembly rate of capsid size . The simulation continues until full depletion of the unfavorable capsid size.
From equilibrium theory and experimental observations, we have to assume that there are some free dimers remaining in the solution before the quench, that is, before the induced shift in and in salt concentration that on the time scale of the experiment is (virtually) instantaneous.
Therefore, we invoke a non-zero value as our initial free dimer concentration. Quickly after starting S 18 the simulation, the dimer concentration converges to a fixed concentration relatively close to what must be the smaller critical concentration. Having initially more dimers in the system leads to the fast formation of capsids. On the other hand, a low dimer concentration at the start of the simulation increases the initial disassembly rate. In order to avoid both of these conditions, we choose the initial dimer concentration near the concentration it converges to. It also helps us to avoid any divergence in the simulation as the dissociation rates increase significantly at low dimer concentrations.
Based on the dimer concentration at the end of the experiment, which is relatively close to the critical concentration of the more stable species, we approximate the total dimer concentration is around 10 times larger than the critical concentration. Therefore, the overall protein in the unfavorable capsid we set at 10 * , where * = ( * 1 , * 3 ). (See table S4 and S5 for parameter values.) Due to the universality of the phenomena, the model is capable of reproducing the experimental results by using different binding energies. We decide to fix the binding energy of T=1 in our framework and generate experimental results only by changing the binding energy of T=3. This allows us to have a better comparison between the two types of experiment.

The fraction of protein dimers in free solution and in capsids
In the main text we show results of the fraction of proteins in the two types of capsid, , as we  Figure S5), implying that the increase in the protein fraction in capsids is not due to the assembly of the free dimers initially present in the solution.
This agrees with what is seen in the experiments, see Figure S7. This supports our suggestion that one capsid size disassembles into free dimers and that these proteins re-assemble into the other capsid size, and that the fraction is the relevant quantity describing the assembly and disassembly kinetics for the problem in hand.

Fitting of the simulation results with respect to different reference points
We have calibrated the simulation results using a reference point in the data series, as mentioned in the caption for figure 2C and 3C. Here we re-calibrated the same simulation data set with respect to a number of reference points to verify the robustness of our fitting procedure. We find that the curve fits depend only relatively weakly on the choice of reference point ( Figure S6).
Unfortunately, our numerical implementation of Classical Nucleation Theory does not allow us to find the fundamental time scale, that is, the time scale associated with the attempt frequency. In spite of this, we are able to show that the disassembly and assembly of the two different capsid sizes can be explained by CNT.

Table of parameters
All parameters related to the simulation of disassembly of T=3 and assembly of T=1, and vice versa, discussed in the main text and used in our comparison with the experiments are tabulated below.